Math 6360 - 11827

Math 6360 - 11827

10-11:30 TTH - Fall 2003

Instructor: Jeff Morgan, 3-3455, jmorgan@math.uh.edu

Course Materials: Class notes and the text Linear Operator Theory in Engineering and Science, by Naylor and Sell, published by Springer-Verlag.

Course Outline:

I	An Introduction to metric spaces, normed linear spaces, bounded linear operators, Banach and Hilbert Spaces, and the contraction mapping theorem.
II	Applications: Well posedness results for initial value problems and existence results for various integral equations.
III	The theory of compact operators, including the Fredholm alternative, the Riesz Schauder theorem, the spectral theorem for compact self-adjoint operators on Hilbert spaces, and the Fredholm splitting theorem.
IV	Applications: p-th order two-point boundary value problems and Fredholm integral equations.
V	A first look at Sobolev spaces (restricted to functions of a single variable): Distributional derivatives, Poincare-type inequalities and embedding results.
VI	Applications: Weak formulation of two-point boundary value problems, asymptotic estimates for eigenvalues of Sturm-Liouville operators, and spanning properties of eigenfunctions of Sturn-Liouville operators.
VII	Fixed point results for nonlinear operators: Schauder fixed point theorem, Leray Schauder fixed point theorem, and monotonicity results.
VII	Applications: Solvability results for nonlinear two-point boundary value problems, constructive techniques using sub and super solutions, and numerical approximation of linear and nonlinear boundary value problems.
VIII	Calculus on Banach Spaces: Gateaux and Frechet derivatives, Newton’s method, the implicit function theorem, bifurcation from a one dimensional kernel, and Hopf bifurcation.
IX	Applications: Multiplicity results for nonlinear two-point boundary value problems and periodic solutions to nonlinear systems of ODEs.

Grades: Homework (50%) and two exams (50%). Note that one of the exams will be a written final.

An Introduction to metric spaces, normed linear spaces, bounded linear operators, Banach and Hilbert Spaces, and the contraction mapping theorem.

Applications: Well posedness results for initial value problems and existence results for various integral equations.

The theory of compact operators, including the Fredholm alternative, the Riesz Schauder theorem, the spectral theorem for compact self-adjoint operators on Hilbert spaces, and the Fredholm splitting theorem.

Applications: p-th order two-point boundary value problems and Fredholm integral equations.

A first look at Sobolev spaces (restricted to functions of a single variable): Distributional derivatives, Poincare-type inequalities and embedding results.

Applications: Weak formulation of two-point boundary value problems, asymptotic estimates for eigenvalues of Sturm-Liouville operators, and spanning properties of eigenfunctions of Sturn-Liouville operators.

Fixed point results for nonlinear operators: Schauder fixed point theorem, Leray Schauder fixed point theorem, and monotonicity results.

Applications: Solvability results for nonlinear two-point boundary value problems, constructive techniques using sub and super solutions, and numerical approximation of linear and nonlinear boundary value problems.

Calculus on Banach Spaces: Gateaux and Frechet derivatives, Newton’s method, the implicit function theorem, bifurcation from a one dimensional kernel, and Hopf bifurcation.

Applications: Multiplicity results for nonlinear two-point boundary value problems and periodic solutions to nonlinear systems of ODEs.